Asymmetric Laplace laws and modeling financial data
type: Item
instance of: scholarly article
title: Asymmetric Laplace laws and modeling financial data
zbMATH Open document ID: 1002.60012
DOI: 10.1016/S0895-7177(01)00114-5
author: Tomasz J. Kozubowski
author: Krzysztof Podgórski
published in: Mathematical and Computer Modelling
publication date: 2002-06-13T00:00:00Z
review text: Let \(v_{p}\) be a geometrically distributed random variable with mean \(1/p,\;p\in (0,1)\). A random vector \(Y\) has an asymmetric Laplace (AL) distribution in \(\mathbb R^{d}\) if and only if geometric compounds \(a_{p}\sum_{j=1}^{v_{p}}(X^{j}+b_{p})\), where i.i.d. random vectors \(X^{i}\) have finite variance, converge in distribution to \(Y\) as \(p\to 0\). If \(X^{i}\) have mean vector \(m\) and covariance matrix \(\Sigma\), then the limiting random vector \(Y\) has the characteristic function \(\Psi(t)=\left[1+{1\over 2}t'\Sigma t-it'm \right]^{-1}\). We use the notation AL\(_{d}(\Sigma,m)\) for the distribution given by the previous characteristic function. Let \(Y\sim\text{AL}_{d}(\Sigma,m)\) and \(X\sim N_{d}(0,\Sigma)\). Let \(Z\) be an exponentially distributed random variable with mean 1, independent of \(X\). Then the following representation holds: \(Y=mZ+Z^{1/2}X\). The main properties of AL laws are presented without proofs. The authors apply the AL model to the currency exchange rates data, showing good fit of this model. AL model is clearly better than normal distribution.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 62E10
Mathematics Subject Classification ID: 91B82
zbMATH DE Number: 1755425
zbMATH Keywords: compound distribution
zbMATH Keywords: currency exchange rates
zbMATH Keywords: heavy tailed modeling
zbMATH Keywords: Laplace distribution
reviewed by: Aleksandr D. Borisenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1755425